Please explain/show how to solve these using factors using decomposition. Not ex

Question

Please explain/show how to solve these using factors using decomposition. Not excel!

Thank you!

7. For each of the 4 following cash flows, find the equivalent present value at time 0. Assume an annual Interest rate of 5%. (Another way to state this question is: how much money must be deposited today into an account earning 5% interest so that the given amounts may be withdrawn in the specified years.) Method factors using decomposition your solution should utilize at least 1 (or more) equal payment, linear gradient, or geometric gradient factor(s) Cash Cash Year YearFlow $200 $250 $300 $350 S30 $250 $250 $250 $250 $250 $250 $250 $1,000 $1,200 $1,440 $1,728 $2,073.6 S1,100 S1,200 1,310 S1,431 $1,564 4 4 $450 $100 $450 $500 $550 10

Explanation / Answer

Equivalent present value at time 0 would be the sum of present value of cash flows.

Present value = CFn/(1+r)n, where CFn is cash flow in period n, r is interest rate, and n is time period

In this formula, 1/(1+r)n is a discount factor, when cash flow occur in period n

For stream of cash flows, Present value is sum of present value of all cash flows

Present value of cash flows = CF0/(1+r)0 + CF1/(1+r)1 + .................. + CFn-1/(1+r)n-1 + CFn/(1+r)n

In the given 4 cases, cash flows are occuring for maximum of 10 years, so we will calculate the discount factors for 10 years

Interest rate = 5%

Discount Factor for year 0 = 1/(1+0.05)0 = 1

Discount Factor for year 1 = 1/(1+0.05)1 = 0.952

Discount Factor for year 2 = 1/(1+0.05)2 = 0.907

Discount Factor for year 3 = 1/(1+0.05)3 = 0.864

Discount Factor for year 4 = 1/(1+0.05)4 = 0.823

Discount Factor for year 5 = 1/(1+0.05)5 = 0.784

Discount Factor for year 6 = 1/(1+0.05)6 = 0.746

Discount Factor for year 7 = 1/(1+0.05)7 = 0.711

Discount Factor for year 8 = 1/(1+0.05)8 = 0.677

Discount Factor for year 9 = 1/(1+0.05)9 = 0.645

Discount Factor for year 10 = 1/(1+0.05)10 = 0.614

Now we will multiply this discount factor with the cash flow for respective cash flows to calculate the present value of that cash flow and sum them to calculate the present value of stream of cash flows

Case a.) Present value = (200*1) + (250*0.952) + (300*0.907) + (350*0.864) + (400*0.823) + (450*0.784) + (100*0.746) + (450*0.711) + (500*0.677) + (550*0.645) + (600*0.614) = 3149.946

Case b.) Present value = (0*1) + (30*0.952) + (250*0.907) + (250*0.864) + (250*0.823) + (250*0.784) + (250*0.746) + (250*0.711) + (250*0.677) = 1406.279

Case c.) Present value = (0*1) + (1000*0.952) + (1200*0.907) + (1440*0.864) + (1728*0.823) + (2073.6*0.784) = 6331.092

Case d.) Present value = (0*1) + (1100*0.952) + (1200*0.907) + (1310*0.864) + (1431*0.823) + (1564*0.784) = 5670.404

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